This invention relates to materials science and, more particularly, to processing of materials using microwave radiation for heating the materials. It currently requires many hours to produce densified articles of alumina and other refractory materials using conventional heat sources. Articles of relatively small size have been sintered using microwave radiation, but sintering of large articles had not been successfully demonstrated. A multi-mode microwave cavity contains an inherently non-uniform electromagnetic field, but smaller bodies are not significantly affected by microwave field non-uniformity. Prior to the present invention, it has not been possible to densify by microwave sintering large articles of these materials having the form of plates or disks, as articles warped, cracked, and shattered, due to the non-uniform field. Hot spots were caused by the areas of concentrated microwave energy.